Invariant dimensions and maximality of geometric monodromy action

TL;DR

This paper investigates the invariant dimensions of l-adic and mod l representations from sheaves on algebraic varieties over finite fields, establishing conditions for maximal geometric monodromy action and providing examples of semisimplicity.

Contribution

It compares invariant dimensions of l-adic and mod l representations and proves a maximality result for geometric monodromy action under semisimplicity and large l.

Findings

Invariant dimensions of V_l and ar V_l are compared.

Maximality of geometric monodromy action is established under certain conditions.

Examples of semisimple ar V_l for large l are provided.

Abstract

Let X be a smooth separated geometrically connected variety over F_q and f:Y-> X a smooth projective morphism. We compare the invariant dimensions of the l-adic representation V_l and the F_l-representation \bar V_l of the geometric \'etale fundamental group of X arising from the sheaves R^wf_*Q_l and R^wf_*Z/lZ respectively. These invariant dimension data is used to deduce a maximality result of the geometric monodromy action on V_l whenever \bar V_l is semisimple and l is sufficiently large. We also provide examples for \bar V_l to be semisimple for l>>0.